Math 347 Worksheet on logical statements A.J. Hildebrand

Logical statements: Practice Problems Solutions

1. Implications: Express each of the following statements as a logical implication (e.g., A ⇐ (¬B)) orequivalence (e.g., A⇔ B). Also state its negation in English (in a form like “A is true, but B is false”).

(a) If A holds, then B holds.A⇒ B. Negation: A is true, and B is false.

(b) A is true only if B is true.A⇒ B. Negation: A is true, and B is false.

(c) A is true whenever B is true.B ⇒ A. Negation: B is true, and A is false.

(d) A is false only if B is false.¬A⇒ ¬B. Negation: A is false, and B is true.

(e) A is a necessary condition for B.B ⇒ A. Negation: B is true, and A is false.

(f) A is necessary and sufficient for B.A⇔ B. Negation: A is false and B is true or A is true and B is false.

(g) A holds if and only if B holds.A⇔ B. Negation: A is false and B is true or A is true and B is false.

2. Negations of English sentences. Negate the following statements. Express the negations in English,avoiding the use of words of negation when possible.

(a) All classroom have at least one chair that is broken.Negation: There exists a classroom in which no chair is broken.

(b) No classroom has only chairs that are not broken.Negation: There exists a classroom that has only good chairs.

(c) Every student in this class has taken Math 231 or Math 241.Negation: There exists a student in this class who has taken neither 231 nor 241.

(d) Every student in this class has taken Math 231 and Math 241.Negation: There exists a student in this class who has taken at most one of the courses 231 and 241 (or “whohas not taken both 231 and 241)”

(e) In every section of Math 347 there is a student who has taken neither Math 231 nor Math 241.Negation: There exists a section of 347 in which every student has taken 231 or 241.

3. Negations of mathematical statements, I. Translate the following sentences into logical notation,negate the statement using logical rules, then translate the negated statement back into English, avoidingthe use of words of negation when possible. (Below f is a function from R to R, and x0 a given real number.)

A bit harder, but very instructive: Many of the statements define familiar properties of functions (e.g.,boundedness, monotonicity, etc.), or negations of such properties. Try to uncover these definitions andexpress in simple language the functions that are described by the statements.

(a) f(x, y) 6= 0 whenever x 6= 0 and y 6= 0.Symbolic notation: (∀x, y ∈ R)[x 6= 0 ∧ y 6= 0⇒ f(x, y) 6= 0]Negation: (∃x, y ∈ R)[x 6= 0 ∧ y 6= 0 ∧ f(x, y) = 0]There exist x 6= 0 and y 6= 0 such that f(x, y) = 0.Remark: Since the variables x and y are not explicitly quantified in the given statement (e.g., by an explicitstatement of the form “for all x ∈ . . . and all y ∈ . . . )”, they are understood to be arbitrary elements in theunderlying universe, i.e., arbitrary real numbers, so the given statement is understood to be interpreted as“For all x, y ∈ R, f(x, y) 6= 0 whenever x 6= 0 and y 6= 0.”Interpretation: The functions f(x, y) defined by this statement are exactly those that are non-zero outsidethe coordinate axis.

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Math 347 Worksheet on logical statements A.J. Hildebrand

(b) For all M ∈ R there exists x ∈ R such that |f(x)| ≥M .Symbolic notation: (∀M ∈ R)(∃x ∈ R)(|f(x)| ≥M).Negation: (∃M ∈ R)(∀x ∈ R)(|f(x)| < M).There exists M ∈ R such that for all x ∈ R we have |f(x)| ≤M .Interpetation: The negation is the definition for “f is bounded”. Hence the original statement is equivalentto “f is not bounded”.

(c) For all M ∈ R there exists x ∈ R such that for all y > x we have f(y) > M .Symbolic notation: (∀M ∈ R)(∃x ∈ R)(∀y > x)(f(y) > M).Negation: (∃M ∈ R)(∀x ∈ R)(∃y > x)(f(y) ≤M).There exists M ∈ R such that for all x ∈ R there exists y > x such that f(y) ≤M .Interpretation: The given statement is the definition of limx→∞ f(x) =∞.

(d) For all x ∈ R there exists y ∈ R such that f(y) > f(x).Symbolic notation: (∀x ∈ R)(∃y ∈ R)(f(y) > f(x)).Negation: (∃x ∈ R)(∀y ∈ R)(f(y) ≤ f(x)).There exists x ∈ R such that for all y ∈ R we have f(y) ≤ f(x).Interpretation: The given statement means that the function f does not have a maximum value; i.e., everyvalue f(x) is “beaten” by some other value f(y). Its negation means that f has a maximum, namely f(x),where x is the number whose existence is guaranteed by the phrase “there exists x ∈ R”.

(e) For every ε > 0 there exists x0 ∈ R such that |f(x)| < ε for all x > x0.Symbolic notation: (∀ε > 0)(∃x0 ∈ R)(∀x > x0)(|f(x)| < ε).Negation: (∃ε > 0)(∀x0 ∈ R)(∃x > x0)(|f(x)| ≥ ε).There exists an ε > 0 such that for all x0 ∈ R there exists an x > x0 such that |f(x)| ≥ ε.Interpretation: The given statement is the definition of limx→∞ f(x) = 0.

(f) For every ε > 0 there exists δ > 0 such that |f(x)− f(x0)| < ε whenever |x− x0| < δ.Symbolic notation: (∀ε > 0)(∃δ > 0)(∀x ∈ R)[|x− x0| < δ ⇒ |f(x)− f(x0)| < ε].Negation: (∃ε > 0)(∀δ > 0)(∃x ∈ R)[|x− x0| < δ ∧ |f(x)− f(x0)| ≥ ε].There exists an ε > 0 such that for all δ > 0 there exists an x such that |x− x0| < δ and |f(x)− f(x0)| ≥ ε.Interpretation: The given statement is the definition for continuity of f at x0.Remark: In the given statement the variable x is not explicitly quantified (i.e., no phrase such as “there existsx ∈ ...” or “for all x ∈ ...”), so x is to be interpreted as an arbitrary element of the underlying universe, i.e.,in the sense of “∀x ∈ R”.

4. Negations of mathematical statements, II. This problem requires the formal definitions of a boundedset or function, and increasing, decreasing, nonincreasing, nondecreasing functions. These definitions canbe found in Chapter 1 of the text and are collected below. (Here S is any set of real numbers, and f denotesa function from R to R.)

• S is bounded if there exists M such that |x| ≤M for all x ∈ S.

• f is bounded if there exists M such that |f(x)| ≤M for all x ∈ R.

• f is increasing (or strictly increasing) if f(x) < f(y) whenever x < y.

• f is nondecreasing (or weakly increasing) if f(x) ≤ f(y) whenever x < y.

• f is decreasing (or strictly decreasing) if f(x) > f(y) whenever x < y.

• f is nonincreasing (or weakly decreasing) if f(x) ≥ f(y) whenever x < y.

(a) Express the statement “f is not bounded” without using words of negation.“(∀M ∈ R)(∃x ∈ R)(|f(x)| > M).”“For all M ∈ R there exists x ∈ R such that |f(x)| > M .”

(b) Express the statement “f is not increasing” (i.e., the negation of the “increasing” property) withoutusing words of negation.

“(∃x, y ∈ R)((x < y) ∧ (f(x) ≥ f(y)))”“There exist real numbers x < y such that f(x) ≥ f(y).”

(c) Compare the definitions of “nonincreasing” and “not increasing” (the latter being the negation of“increasing”). Does one imply the other? Are there functions that satisfy one property, but not theother?

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Math 347 Worksheet on logical statements A.J. Hildebrand

“Not increasing” means that there exist real numbers x < y for which the inequality f(x) ≥ f(y) holds.“Nonincreasing” means that this inequality holds for all real numbers x < y. Thus, “nonincreasing” is astronger property than “not increasing”, and hence implies the latter. A nonincreasing function is always notincreasing, but the converse is in general not true. For example, the function f(x) = sinx is not increasing,but also not nonincreasing.

5. Practice with epsilon-delta definitions. Recall the epsilon-delta definition of “limx→0 f(x) = 0”:

(*) “For every ε > 0 there exists δ > 0 such that |f(x)| < ε whenever |x| < δ.”

The following statements are small perturbations of this definitions, some of which are equivalent to theoriginal definition, while others are “botched” versions of this definition that have a drastically differentmeaning.

Which versions are equivalent to the continuity definition, and which are not?

Harder, but very instructive: For those definitions that are not equivalent to limx→0 f(x) = 0, try todetermine, in as simple a language as possible, what they really define. Find examples (if they exist) offunctions that satisfy the definition, and of functions that don’t satisfy it. (Cf. Exercises 2.25–2.27 in thetext for similar problems.)

(a) For every ε > 0 there exists δ > 0 such that for all x ∈ R, |x| < δ implies |f(x)| < ε.Analysis: This is the same as (*), except that “|f(x)| < ε whenever |x| < δ” is replaced by “|x| < δ implies|f(x)| < ε.” Since the latter statements are equivalent, the given statement is equivalent to the original one,(*).

(b) For every δ > 0 there exists ε > 0 such that for all x ∈ R, |x| < δ implies |f(x)| < ε.Analysis: This differs from (*) in that the types of quantifiers (∀ or ∃) for δ and ε have been switched. This,however, changes the meaning completely, since now δ is the given quantity, and ε can be chosen freely. Thestatement is no longer equivalent to that in (*). For example, any constant function f(x) = c satisfies the givenstatement since choosing ε to be a number greater than |c| (for example, ε = |c|+ 1) guarantees |f(x)| < ε.

(c) There exists δ > 0 such that for every ε > 0 and for all x ∈ R, |x| < δ implies |f(x)| < ε.Analysis: This differs from (*) in that the order of the quantifiers ∀ε > 0 and ∃δ > 0 has been switched. Thisagain changes the meaning completely, and the statement is not equivalent to (*). (For example, the functionf(x) = x satisfies (*), but does not satisfy the given statement.)

(d) For every ε > 0 and for all x ∈ R there exists δ > 0 such that |x| < δ implies |f(x)| < ε.Analysis: Here the order of the quantifiers ∀x ∈ R and ∃δ > 0 has been switched. With δ as the inside variableand x as the outside variable, δ can depend on x (in addition to depending on ε) and thus be “customized”to any given x-value. This alters the situation completely.

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